Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,11)(5,8,13,12)(10,14,15,16)(17,18,22,23)(19,20,25,24)(21,29,30,26)(27,31,32,28)(33,34)(35,36)(37,45)(38,40)(39,43)(41,46)(42,47)(44,48);(2,4)(7,11)(8,12)(14,16)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,31)(30,32)(33,35)(34,39)(36,40)(37,42)(38,43)(41,47)(44,45)(46,48);(3,9)(5,13)(7,11)(8,12)(18,23)(19,25)(21,30)(28,31)(34,40)(36,39)(37,46)(42,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,29)(30,32)(33,36,38,39)(34,35,40,43)(37,44,46,47)(41,42,45,48);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,47)(46,48);(1,5,6,13)(2,8,4,12)(3,10,9,15)(7,14,11,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,37,38,46)(34,41,40,45)(35,42,43,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=gh,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837456);